Galaxy clusters are the largest and most massive gravitationally bound systems in the universe. They form on the densest points of the cosmic web and hence contain a wealth of information about structure formation. Specifically, cluster abundance as a function of mass and redshiftis sensitive to cosmological parameters that govern the geometry and structure growth in the universe. Of further importance is the different degeneracydirections between structure growth parameters probed by clusters compared to cosmic microwave background (CMB) or baryonic acoustic oscillations, which provide compelling joint constraints.This has been demonstrated previously in the literature (for example, Mantz et al. 2008;Vikhlinin et al. 2009;Rozo et al. 2010;von der Linden et al. 2014;de Haan etal. 2016;Salvati et al. 2018;Bocquet et al. 2019;Zubeldia & Challinor 2019;Planck Collaboration et al. 2020a), and the potential of clustersas cosmological probes from future surveys has also been a subject of extensive study (for example, Holder et al. 2001;Lima & Hu 2004;Sartoris et al. 2012;Mak &Pierpaoli 2013, andrecently Louis &Alonso 2017;Madhavacheril et al. 2017;Cromer et al. 2019;Gupta et al.2020).
Hot electrons in the intracluster medium (ICM) transfer energy to CMB photons through inverse Compton scattering (Sunyaev & Zel'dovich 1970). This thermal Sunyaev-Zeldovich effect (tSZ) has been used to detect clusters from CMB surveys (Bleem et al. 2015;Planck Collaboration et al. 2016a;Hilton et al. 2018Hilton et al. , 2021;;Huang et al. 2020;Bleem et al. 2020), and the number of clusters has been rapidly growing from a few hundreds to thousands with the increase in sensitivity of the CMB surveys (Benson et al. 2014;Henderson et al. 2016;Bender etal. 2018).Future CMB surveys like CMB-HD (Sehgal et al. 2019),CMB-S4 (CMB-S4 Collaboration 2019), Cosmic Origin Explorer (CORE; Melin et al. 2018b),and Simons Observatory (SO; Ade et al. 2019) will increase the sample size several fold, producing mass-limited cluster catalogsdown to M 500c 10 14 M e . In the context of galaxy clusters, future CMB surveys play an important and unique role as they open the window to the high redshift z 2 universe using the redshift independent tSZ effect, enabling the detection of distantclusters. These distant clusters willotherwise be hard to detect using optical or X-ray surveys, and as a result, future tSZselected cluster samples will be complementary to the ones from Large Synoptic Survey Telescope (LSST) at the Vera C.Rubin
Observatory (LSST Science Collaboration et al. 2009),Euclid (Laureijs et al. 2011), and eROSITA (Merloni et al. 2012).
The clusters also gravitationally lens the background CMB, an effectknown as CMB-cluster lensing. After the first set of detections using the CMB temperature by the Atacama Cosmology Telescope (ACT;Madhavacherilet al. 2015), South Pole Telescope (SPT; Baxter et al. 2015), and Planck (Melin & Bartlett 2015;Planck Collaboration et al. 2016b), the field has rapidly evolved to use the signal to calibrate richnessmass scaling relations of optically selected galaxy clusters (e.g., Geach & Peacock 2017) and to warrant the first polarizationonly detection of the signalby SPTpol survey (Raghunathan et al. 2019b). Like the tSZ effect, CMB-cluster lensing also plays a key role in facilitating the mass measurements of distant clusters expected from future CMB surveys. This is difficult with galaxy weak lensing since the signal-to-noise ratio (S/N) of lensed background galaxiesdrops significantly at high redshifts.Planck Collaboration et al. (2016b) and Zubeldia & Challinor (2019) used CMB-cluster lensing information to derive cosmological constraints with the Planck cluster sample, while Alonso et al. (2016) and Madhavacheril et al. (2017) studied the potentialof CMB-cluster lensing eitherindependently or in combination with galaxy weak lensing to calibrate the observable-mass ( Y M SZ -
) scaling relations of clusters from CMB-S4 and its impact on cluster cosmology.While extensive studieshighlighting the importance of clustersas cosmologicalprobes existin the literature, understanding the virialization mechanism and astrophysicsof high redshift (z 2) clusters mostly remains an unexplored territory owing to the lack of observations.
In this work, we focus on astrophysical and cosmological constraints using cluster samples from future CMB surveys. Our primary focus is on the wide area survey (S4-Wide) of the CMB-S4 experiment, but we also provide predictions for the smaller but deeper CMB-S4 de-lensing survey (S4-Ultra deep). The proposed CMB-HD experiment is also added to the list as an ideal case. We startby forecasting the number of tSZ-selected galaxy clusters from the three surveys. The simulations used for forecasting are designed to capture most of the effects expected in a real survey. They contain atmospheric and instrumental noise along with signals from galactic and astrophysical foregrounds. The detected clusters are then binned in lensing mass (obtained using CMBcluster lensing), tSZ S/N q, and redshift. We combine the binned cluster counts N(z, M L , q) with primary CMB temperature and polarization spectra to derive parameter constraints. In addition to cosmology and the Y M SZ -scaling relation, we also study the evolution of the ICM using high redshift clusters. For this, we modify the tSZ signals of clusters, using two parameterizations of the virialization model ( )
where Y th and Y nth are the thermaland nonthermal components, respectively, of the total integrated Compton-y signal Y tot = Y th + Y nth . In the first approach, we use a linear model to scale the tSZ signals from clusters with z 2. We note that this toy model with a step function atz 2 is over-simplistic for accurately capturing the redshift dependence of the cluster virialization process.For example, Fakhouri et al. (2010) showed that mergers, which are considered an important source of nonthermal pressure, increase as a function of redshift, which would modify the virialization mechanism of high redshift clusters. To take this into account, we build a second more realistic model using a fitting formalism, ( )
, that has been derived using the analytic modelof the nonthermal pressure in the ICM (Shi & Komatsu 2014;Green etal. 2020) and tested using Omega500 simulations (Nelson et al. 2014a;Shi et al. 2015).
This paper is structured as follows: We describe the simulation components, cluster virialization model, detection algorithm, mass calibration using CMB lensing, and the Fisher formalism to combine binned cluster counts with primary CMB information in Section 2.In Sections 3.1 and 3.2, we discuss the baseline results including cluster detection sensitivity, survey completeness, and cluster counts. The modification to cluster sensitivity and counts due to changes in the virialization mechanism are given in Section 3.3.We discuss the Fisher forecasts along with the impact of several choices we make in Sections 3.4 and 3.5. We test the effect of cluster correlated foreground signals in Section 3.6 and finally conclude in Section 4.
Throughout this work, we use Planck 2015 cosmology (TT+ lowP in Table 4 of Planck Collaboration etal. 2016c) and report cluster masses in units of M 500c , which is the mass within a sphere of radius R 500c where the density is 500 times the critical density of the universe at the cluster redshift.
The simulations used for this study are 2° × 2° wide CMB temperature realizations with a pixel resolution of 0 5. Other than primary CMB, the simulations also contain the following frequency-dependent signals: cluster tSZ, galactic and astrophysical foregrounds,and experimental noise (both atmospheric and instrumental). The underlying power spectrum used to generate the primary CMB is the large-scale structure lensed temperature spectrum C ℓ TT for the fiducial Planck 2015 cosmology (Planck Collaboration et al. 2016c) obtained using CAMB (Lewis etal. 2000) software.The cluster tSZ signal is modeled using a generalized Navarro-Frenk-White (NFW; Navarro etal. 1996;Zhao 1996;Nagai et al. 2007a;Arnaud et al. 2010) profile as described below in Section 2.4. Galactic and astrophysicalforeground modelings are presentedin Section 2.3. The simulated maps are then convolved by experimentalbeam functions,assumed to be Gaussian (see Table 1). Finally, we add noise realizations to the simulated maps. We model the noise spectra to include both atmospheric and instrumental noise as (Tegmark 1997)
where T 2 D corresponds to detector white noise while ℓ knee and α knee are used to model the atmospheric1/f noise. This parameterization gives us a sense of the range of multipoles being affected by the atmospheric (ℓ < ℓ knee ) and instrumental noise components (ℓ ℓ knee ).
We consider three future CMB surveys in this work: CMB-HD, S4-Wide, and S4-Ultra deep. and experimentalbeams of each frequency band for the three surveys. CMB-HD is a proposed high-resolution millimeter-wave survey scanning large regions of the sky from Chile with a 30 m primary mirror and designed to operate in seven bands from 30-350 GHz (Sehgal et al. 2019(Sehgal et al. ,2020)).CMB-S4 is an upcoming survey that is currently in its design stages and expected to start operations later this decade (CMB-S4 Collaboration 2019). In this work, we only consider the two CMB-S4 large aperture telescope surveys and not the small aperture telescope survey that is aimed at the inflationary gravitational waves.The two S4-LAT surveys (S4-Wide and S4-Ultra deep)will be performed using 6 m class telescopes in sixfoot_1 frequency bands from 30 to 270 GHz. S4-Wide is a legacy survey from Chile and will cover roughly 67% of the sky area. S4-Ultra deep is the "de-lensing" survey that is aiming to provide deep observations of ∼3% of the sky from the South Pole. The primary objective of the S4-Ultra deep survey is to generate high-resolution maps of the dark matter distribution in the universe to facilitate the detection of inflationary B-modes by cleaning the lensing-induced B-modes. However, given the large telescope size, S4-Ultra deep also has the capability to detect high redshift SZ clusters, as we show in this work. The parameters governing the atmospheric 1/f noise, ℓ knee and α knee , are listed in Table 2 (CMB-S4 Collaboration 2019). For simplicity, we assume the 1/fmodel and sky fraction for CMB-HD to be the same as those in the Chilebased S4-Wide survey.
Although extragalactic foregrounds, emissions from dusty star-forming galaxies (DSFGs) in particular, are expected to be the major source of contamination for cluster detection,the footprint of Chile-based experiments cover 67% of the sky area and will be subjected to contamination from galactic emission. Hence, we consider both galactic and extragalactic foreground signals for CMB-HD and S4-Wide. For S4-Ultra deep, we only include extragalactic foregrounds, as it will observe a relatively clean region of the sky shown as the yellow dashed contours in Figure 1.
The galactic foreground signals, dust and synchrotron, are position dependentand hence one cannotrely on Gaussian realizations of an underlying power spectrum for the entire footprint. To this end, we use the publicly available pySM3 dust and synchrotron map simulations,foot_2 which were built specifically in the context of CMB-S4. For more details about pySM3 simulations,we refer the reader to the originalwork (Thorne et al.2017),which is partly based on the Planck Sky Model code (Delabrouille et al. 2013). We use S0_d0 dust and S0_s0 synchrotron models of pySM where the dust temperature, dust emissivity index, and synchrotron spectral index do not have spatialvariations.The models also ignore any non-Gaussianities. Other galactic signals like free-free and anomalous microwave emissions, which should have a negligible impact on cluster searches, are ignored in this work. To estimate the position-dependent galactic foregrounds, we first divide the S4-Wide footprint into two high and low emission regions, shown as the red and blue contours in Figure 1. The high emission region corresponds to ±15° and encompasses most of the signals from the galactic plane. The low emission region corresponds to regions in the range -45° b -30°. We compute the temperature power spectra C ℓ gal of dust and synchrotron signalsof both these regions for all of the frequency bands of interest.
We assume 23% ( f sky = 17%) of the S4-Wide footprint to have galactic signals similar to the high emission region and the galactic signals in the remaining 77% ( f sky = 50%) to be similar to the low emission region. As is evident from the figure, the blue contours do not correspond to the cleanest region in the S4-Wide footprint, and hence this is a conservative choice. With this assumption, we use the C ℓ gal spectra estimated in the two regions to generate Gaussian realizations of galactic emission and add them to our maps to produce two sets of simulated skies.The underlying power spectrum for the other signals in the two sets is the same. By doing this, we approximate the galactic power spectra to be constant inside the two regions. We validated this assumption by dividing the S4-Wide footprint into six regions (latitude steps of Δb = 15°) with different galactic emission and do not find a significant difference in the number of detected clusters between the two approaches. We follow the same approach for CMB-HD. Like mentioned above,since the footprint of S4-Ultra deep lies in a relatively clean region, we do not include galactic foregrounds for S4-Ultra deep simulations.
The CMB-S4 pySM3 simulations do not include the 350 GHz band,which is required for CMB-HD. We obtain auto-spectra of the galactic dust at the 350 GHz band and its cross correlation with other bands by simply scaling the
where ν 0 = 270 GHz and ν 1 , ν 2 ä [30,40,90,150,220,270,353] GHz. The terms ò ν and η ν in Equation (2) when combined represent the spectral energy distribution ( f ν , SED) of dust, and we use a modified blackbody of the form ).We ignore synchrotron signals at 350 GHz since they are expected to be negligible compared to dust at such high frequencies.
Diffuse extragalactic foreground signal can be decomposed into: emissions from DSFGs and radio galaxies (RGs) below the detection threshold; and kinetic SZ (kSZ) and tSZ signals. Note that DSFGs are responsible forcosmic infrared background (CIB) anisotropies at millimeter/submillimeter wavelengths, and we sometimes use the two terms, DSFGs and CIB, interchangeably in this work. DSFG and RG signals were all modeled as Gaussian realizations using SPT power spectra measurements (George et al. 2015;Reichardt et al. 2021). SPT observationsmasked DSFGsand RGs detected above5σ, which correspondsto a flux threshold of S 150 ∼ 6 mJy. However, the 5σ detection limit for the future surveys considered here will be much lower:S 150 ∼ 2 mJy for CMB-S4 (CMB-S4 Collaboration 2019) and 0.1 mJy for CMB-HD (Sehgal et al. 2019). For CMB-S4, we do not modify the masking threshold and simply use SPT measurements. Thus, DSFG and RG signals injected into S4-Wide and S4-Ultra deep simulations are conservative estimates. For CMB-HD, however, Sehgalet al. ( 2019) claim thatsources with flux above S 150 0.04 mJy can be efficiently removed by detecting them at 3σ in the 270/350 GHz bands.This lowers the DSFG power in 150 GHz by ×17, and we adoptthat strategy here. The masking threshold for RG is not modified from SPT values. The frequency dependence of DSFG and RG signals is also adopted from SPT (George et al. 2015;Reichardtet al. 2021) measurements. We introduce decorrelation in the DSFG signals between the 270/350 GHz and 150 GHz bands using SPT × Herschel/Spectral and Photometric Imaging Receiver (SPIRE) measurements (Viero et al. 2019). We estimate the correlation coefficient between the 150 GHz and 270/350 GHz bands by interpolating the values in Table 1 ofViero
and has no frequency dependence. For the diffuse tSZ,we consider power from all haloes with M 500c 10 13 M e , modeled using the Arnaud profile (Arnaud et al. 2010), in the redshift range z ä [0.1, 4.0]. Both the diffuse kSZ and tSZ signals are simulated as Gaussian realizations using the respective power spectra described above.
In our fiducial setup, DSFG/RG/kSZ signals are assumed to be uncorrelated to the cluster under study. This is, however, not entirely correct as galaxies preferentially reside inside clusters, and the cluster motion can also give rise to kSZ signals. We test this assumption in Section 3.6 by injecting cluster correlated foreground signals using Websky (Stein et al. 2020) and MultiDark Planck 2 (MDPL2,Y. Omori 2022, in preparation) simulations.
Figure 1. Map of the galactic dust emission at 150 GHz from pySM3 simulations. The expected footprints for S4-Wide ( f sky = 67%) and S4-Ultra deep ( f sky = 3%) are highlighted in black and yellow. Regions of high and low galactic emissions used for injecting galactic foregrounds in our simulations are marked as red and blu contours. We assume the galactic emission in 23% of the S4-Wide footprint ( f sky = 17%) to be similar to high galactic emission in the red contours and the emission in the remaining 77% ( f sky = 50%) to be similar to low galactic emission in the blue contours (marked as baseline). This is a conservative choice as the blue contours ar not the cleanest region in the S4-Wide footprint. We use the same S4-Wide footprint and strategy for CMB-HD. Since S4-Ultra deep will observe in a relatively clea patch,we do not include galactic foregrounds for S4-Ultra deep simulations.
We model the ICM pressure using the dimensionless universal pressureprofile P e (x) proposed by Nagai et al. (2007a) where the distance to the cluster center x ≡ xR 500 , expressed in terms of virial radius R 500 , and concentration parameter c 500 are related to scale radius r s as x = r/r s and c 500 = R 500 /r s . The best-fit values of the parametersare (Arnaud et al. 2010
where σ T in the Thomson cross section, c is the velocity of light, m e is the electron mass,T CMB = 2.73 K is the mean temperature of the CMB, and g SZ (ν) is the frequency dependence of the tSZ signal, which, ignoring relativistic SZ corrections (e.g., Itoh et al. 1998;Chluba et al. 2012), is given by
where h and k B are Planck and Boltzmann constants, respectively. We integrate y(x) over the angular extent of the cluster R 500 to obtain the total integrated clusterCompton Y SZ c 500 defined using Planck Collaboration et al. (2016b) but generalized based on Alonso et al. (2016) and Madhavacheril et al. (2017) to include mass and redshift evolution as
where M * = 6 × 10 14 M e is the pivotal mass, D A (z) is the angular diameter distance to the cluster at redshift z, E(z) = H(z)/H 0 is the Hubble function,and M 500c is the mass of the cluster. v(z) in the above equation is the cluster virialization model adopted to modify the cluster tSZ signal and is explained below in Section 2.5.
with the fiducial values set to 0.127 Y log ,0 s = , α σ = 0, and γ σ = 0 (Louis & Alonso 2017).
Not much is known aboutthe astrophysics of high redshift clusters owing to the lack of sufficient observations. Lately, Mantz et al. (2014) and Mantz etal. (2018) with M 500c ∼ 1 -2 × 10 14 M e that was detected by the X-ray XMM-Newton satellite. Mantz et al. (2018) report that the properties of this distant cluster are in reasonable agreement with the extrapolated scaling relations confirming self-similar evolution of clusters outto z ∼ 2. However, the authors also caution the readers about generalizing the result from a single z ∼ 2 cluster to all high redshift clusters. As we will see later in Section 3.2, CMB-HD and CMB-S4 have the capability to detect hundredsto thousandsof clusters with M 500c 10 14 M e at z 2. Subsequently, we aim to study the physics of the ICM and its evolution out to high redshifts with these potential detections. To this end, we tweak the first term in Equation ( 9
that controls cluster virialization and as a result modifies the cluster tSZ signal. We model v(z) in two different ways as described below.
In the first approach, we use a simple model
where b HSE is the hydrostatic equilibrium (HSE) mass bias set to b 0.2 HSE = (Zubeldia & Challinor 2019;Makiya et al. 2020) and assumed to be constant for clusters at all redshifts. η v (z) is the virialization efficiency of clustersthat modifies the tSZ signal of clusters using a linear scaling as 2016b) except for the introduction of η v (z) for high-z clusters. The fiducial value of η v (z) = 1 for all clusters.
Since the step function at z 2 used in the above model is highly simplistic, we now build a realistic model to parameterizethe redshift dependenceof the virialization process (e.g., Fakhouri et al. 2010) more accurately.In this second approach, we use the analytic model for modeling the evolution of the nonthermal pressurefraction through the cluster assembly and virialization processes (Shi et al. 2015) and their impact on the Y M SZ -relation of high redshift clusters using the model presented in Green et al. (2020).We summarize the modeling and results in the Appendix, in which we use a fitting formalism
= + + derived from the analytical model of nonthermal pressure (Shi & Komatsu 2014;Green et al. 2020) and tested using the Omega500 hydrodynamical cosmologicalsimulation (Nelson et al. 2014a;Shi et al. 2015). We set the fiducial values of the parameters to be A v = 0.155 and B v = 0.189.
We combine the simulated maps in different frequency channels N ch optimally using an internal linear combination (ILC) algorithm and create a Compton-y map as
where the multipole-dependent weights w ℓ for each frequency channelare computed using the SMICA (Spectral Matching IndependentComponentAnalysis) algorithm (Cardoso etal. 2008;Remazeilleset al. 2011;Planck Collaboration et al. 2014) as
The matrix C ℓ has a dimension N ch × N ch and contains the covariance between simulated maps in multiple frequencies at a given multipole ℓ. The frequency response vector a = [-5.33, -5.23, -4.36, -2.61, 0.09, 2.27, 5.95] contains the tSZ spectrum given in Equation ( 8) for [30,40,90,150,220,270,350] GHz channels.The weights in Equation ( 15) for each band are chosen optimally to produce a minimum variance Compton-y map by jointly minimizing the contamination from the noise and foreground signals that are uncorrelated with the cluster.We do not explicitly null any foreground components using a constrained ILC technique (Remazeilles et al. 2011), but study the effect of cluster correlated foreground signals in Section 3.6.
The resultant ILC Compton-y map is then used to compute the S/N of the cluster tSZ signal using a maximum likelihoodbased approach. For blind cluster searches, however,adopting a multiband matched-filtering technique (Melin etal. 2006) would be computationally more feasible as done traditionally in cluster finding using CMB surveys (e.g., Bleem et al. 2015). The two approaches are equivalent. In addition to the cluster tSZ signal, this map includes variance from diffuse tSZ and also the residualCMB, foreground signals, and noise.Using this 2° × 2° ILC y map,we calculate
where y i ≡ y i (θ) is the azimuthally averaged profile of the Compton-y signalin bins i of Δθ = 0 5, out to a maximum 2 max q = ¢ . The chosen max q encompasses the tSZ signal from the majority of the clusters at all redshifts and hence maximizes the S/N. Specifically, c max 500 q q for clusters with z 0.5
where θ 500c = R 500c /D A (z) and D A (z). The measured cluster Compton-y from a cluster with a given mass and redshift is ŷ. We compute theory modelsyth for different massesat the cluster redshift using Equation ( 9) and fit them to the measured ŷ signal.The covariance matrix Ĉ includes contribution from other sources of variance described above. It is computed using N = 2500 simulations as
We use the distribution of best fits recovered from 100 simulationsand computethe S/N as the inverseof the 1σ uncertainty defined by the 16% to 84% confidence range. For each survey, we compute an S/N look-up table in this way for different clusters in an (M 500c , z) grid:
and z ä [0.1,3] with Δz = 0.1. This S/N look-up table is used to select clustersabove the detection threshold S/N ≡ q = 5 in later sections.
We perform internal mass calibration of clusters using their gravitational lensing signatures on both CMB temperature and polarization anisotropies.Cluster kSZ and tSZ signals are expected to introduce significant bias to temperature-based lensing reconstruction (Raghunathan et al. 2017). We mitigate them by employing an inpainted-gradient (Raghunathan et al. 2019a) quadratic temperature lensing estimator (QE; Hu et al. 2007). This estimatorreconstructs lensing using the lensinginduced correlations between a large-scale and a small-scale temperature anisotropies map. Cluster SZ signals, in addition to lensing, can also introduce such correlations, which tend to bias best-fit lensing masses.In the inpainted-gradientQE, we remove SZ signals in the large-scale gradient map by estimating the pixel values at the cluster location using information from adjacent pixels. For polarization,we use the optimal maximum likelihood estimator (MLE; Raghunathan et al. 2017), which reconstructs cluster massesusing the lensing-induced changes to a pixel-pixel covariance matrix. We ignore the covariance between temperature and polarization but note that it can slightly degrade the lensing S/N.
We use the Fisher matrix formalism (Holder et al. 2001) and compute
where θ i , θ j are the astrophysical or cosmological parameters to be constrained; N(z, M L , q) is the number of clusters in a given lensing mass M L , tSZ S/N q, and redshift z bin; and 1/N(z, M L , q) gives the Poisson error in each bin. The summation indices lqz run over the M L , q, and z bins described below in Section 2.8.2. Cluster number counts in a given bin(
where n(M 500c , z true ) ≡ n(M, z true ) is the Tinker et s are obtained using the MLE approach described above,and the errors in lensing mass M L s are determined using CMB temperature and polarization-based reconstructions.
We solve the above integral using a Monte Carlo (MC) sampling approach to estimate N(z, M L , q) and its derivatives ∂N(z, M L , q)/∂θ as a function of the parameter under consideration θ.We start by getting the number of haloes n(M 500c , z) in the following mass and redshiftbins using the Tinker et al. (2008) HMF: M 500c ä [10 13 , 10 16 ] M e with ΔM 500c = 10 12 M e and 0.1 z 3with Δz = 0.1. While statistical uncertaintiesin the HMF parameters could be potentially important (Artis et al. 2021), we defer their impact on the results to a future work. For each halo, we assign a tSZ flux
. The tSZ S/N for the halo is obtained by interpolating the S/N look-up table in Section 2.6.1.Lensing mass and redshifts are also assigned using the distributions
. Next we bin the haloes in lensing mass, S/N, and redshift to obtain binned cluster counts N(z, M L , q) as described below.We repeatthe MC sampling approach 100 times to ensure the convergence of cluster counts N(z, M L , q).
We choose 40 and 25 logarithmic bins for lensing mass and tSZ S/N: M L ä [10 12 , 10 16 ] M e and q ä [5, 500]. For redshift, we consider four different binning schemes. In the baseline case, we use Δz = 0.1 for 0.1 z < 1.5 and conservatively group all high redshift clusters 1.5 z 3 in one massive redshift bin similar to that in Madhavacheril et al. (2017). This is due to the difficulties that will be encountered in measuring redshifts of distant clusters. While dedicated follow-up observations are needed to obtain redshifts for clusters at z 1.5, the absence of an associated signal in multiple LSST bands will still allow us to set a lower limit on the clusterredshifts,and we set this threshold to be z = 1.5. Redshifts of clusters with z < 1.5 can be obtained using upcoming optical and X-ray surveys (LSST Science Collaboration et al. 2009;Merloni et al. 2012).We also explore other choices for redshift binning: (i) an extremely optimistic case of Δz = 0.1 for all clusters;(ii) a less conservativechoice of Δz = 0.1 for 0.1 z < 2 and Δz = 1 for 2 z 3; and (iii) a pessimistic setting by ignoring clusters at z > 1.5. 2.8.3.Derivatives ∂z,M L , q/∂θ We estimate derivatives of binned cluster counts ∂N(z, M L , q)/∂θ as a function of parameterθ using a finite difference method.For this,the MC sampling approach must be repeated twice for every parameter perturbing θ → θ ± ò θ . The randomness in sampling, however, can lead to unstable derivatives, and we avoid this by only estimating counts N(M L , q, z) at the fiducial values of the parameters. For derivatives,we assign weights to haloes based on the ratio of the PDF at the sampled point before and after modifying the parameter values. Subsequently, the weights are decomposed into w Poi , w tSZ , w q , w M L , and w z with the final weight being the product of all of the individual ones. Here,
quantifies the change in number of haloeswhen parameterθ is modified. The other weights (w SZ , w q , w M L , and w z ) are simply the ratio of respective individual PDFs at the sampled point Along with cluster counts, we also make use of the information from primary CMB temperature and polarization power spectra. Since clusters can lens the background CMB, cluster counts will have a nonzero covariancewith CMB lensing power spectrum. We make a conservative choice and fully ignore information from CMB lensing power spectrum in this work. We use lensed CMB spectra but do not correctfor the lensing-inducedcorrelations. Because of the nonzero covariance between clusters and CMB lensing, we note that this can underestimatethe error bars (Green et al. 2017). However, the effect is small, and hence we do not consider it. In a similar vein, we ignore information from the tSZ power spectrum since it must be highly correlated with cluster counts.
We compute CMB Fishermatrices using TT,EE, and TE power spectra ( ℓ 5000 max =
) obtained using CAMB (Lewis et al. 2000) software for the fiducial Planck 2015 cosmology described in Section 2.1.The CMB TT, EE, and TE information comes from the same experiment under consideration. Although we could include Planck information on large scales and in the regions notcovered by the experiments in this work, we avoid them in the baseline setup.We also avoid adding S4-Wide information in the regions not covered by S4-Ultra deep. Like in the case of Compton-y maps, we optimally combine information from all frequency channels using the ILC algorithm to compute the residualnoise (see Table 1 andTable 2)and foreground spectra (see Section 2.3) in the CMB maps for all three surveys. To generatepolarized foregrounds,we assume2% (3%) polarization fractionsfor dusty star-forming (radio) galaxies consistent with measurements from ACT (Datta et al. 2019) and SPT (Gupta et al. 2019). Diffuse kSZ and tSZ signals are assumed to be unpolarized. Information about polarized galactic dust and synchrotron signals comes from pySM3 simulations (see Section 2.3.1).
Figure 2 shows the residual power in the ILC Compton-y maps ( N ℓ yy ) for CMB-HD (yellow), S4-Wide (green), and S4-Ultra deep (red).Solid lines in the figure correspond to noise estimates when galactic emission is not included. CMB-HD and S4-Wide experiments are expected to scan large sky areas ( f sky = 67%), and it is unrealistic to ignore galactic emission. Hence for CMB-HD (yellow) and S4-Wide (green)experiments, we also show the noise curves in regions of low (dashed-dotted)and high (dashed) galactic emission as discussed in Section 2.3.1. Since S4-Ultra deep will observe a region with negligible galactic foregrounds (CMB-S4 Collaboration 2019), we only show a solid red line.
In the absence of galactic emission (solid curves), we find the noise level in CMB-HD maps to be much lower than both S4-Wide and S4-Ultra deep surveys. This is primarily due to the reduced level of CIB signals expected in CMB-HD compared to CMB-S4. As described in Section 2.3.2,note that the CIB power at 150 GHz for CMB-HD is lower than CMB-S4 by 17× (Sehgalet al. 2019). The Compton-y maps from CMB-S4 are fully dominated by residual CIB signals on small scales. Residual CIB signals in Compton-y maps can be lowered by nulling CIB signals assuming one or more spectral energy distributions with a constrained ILC (Madhavacheril et al. 2020) or using partial ILC techniques (Bleem et al. 2022). This CIB reduction comes at the cost of higher noise depending on the choice of cleaning.We ignore this here butstudy the systematicsin the recovered cluster tSZ signals due to emissions from DSFGs within clusters (tSZ × CIB) in Section 3.6.
On large scales, we note a change in noise trend and find noise in CMB-HD to be slightly higher than S4-Ultra deep. This is due to a higher atmospheric noise in CMB-HD, as it will be located in Chile (Sehgal et al. 2020) compared to S4-Ultra deep,which will be observing from the South Pole. For S4-Wide, both atmospheric noise and residual CMB signals dominate the large-scale noise, which is much higher than both CMB-HD and S4-Ultra deep surveys.Including information from Planck will improve the noise performanceon large scales,but we ignore that as we are primarily interested in ℓ 3000 for cluster detection.
When galactic emission is included, as expected, the noise increases forboth CMB-HD and S4-Wide surveys.For S4-Wide, adding low levels of galactic emission (blue contour in Figure 1) only affects large-scale noise (green dashed-dotted line) as small scales are dominated by residual CIB emission. When looking right through the galactic plane (red contour in Figure 1),residual noise (green dashed) is much higher on all scales. For CMB-HD, any level of galactic emission leads to an increased noise on all scales.
For reference,in the gray band, we show the fiducial tSZ power spectrum along with 1σ, 2σ errors from SPT measurement (George et al. 2015). Comparing the gray band with noise curves, we note that all three surveys can map the peak of the tSZ power spectrum (3000 ℓ 4500) with S/N 1 (CMB-S4 Collaboration 2019).
In addition to the instrumental noise and foregrounds, another source of noise for cluster detection is the confusion noise arising due to the diffuse tSZ signal. Note that the noise curve N ℓ yy is much lower than tSZ power spectrum for CMB-HD. While this indicates a high S/N measurement of the tSZ power spectrum on all scales, it limits the sensitivity of cluster detection due to the tSZ confusion noise.
Our baseline results with no modifications to the cluster tSZ signal are presented in Figures 345and Tables 3 and4.
Figure 3 shows the redshift dependence of the minimum cluster mass required to satisfy the detection threshold criterion S/N 5. For reference, we also mark the clusters detected at S/N 4.5 from currentsurveys:ACT (Hilton et al. 2018(Hilton et al. , 2021) ) as blue diamond,Planck (Planck Collaboration etal. 2016b) as red squares, and SPT (Bleem et al. 2015;Huang et al. 2020;Bleem et al. 2020) as black circles. We present two curves for CMB-HD (yellow) and S4-Wide (green): dashed-dotted and dashed curves correspond to sensitivity in regions of low and high levels of galactic emissions, respectively. For S4-Ultra deep, since we do not inject any galactic emission, we only show the solid line containing no galactic foregrounds.
As expected, based on the intuition from Figure 2, minimum detectable cluster mass is lowest for CMB-HD followed by the S4-Ultra deep and S4-Wide surveys. The dominantsource of variance for CMB-S4 surveys comes from the residual CIB contamination present in the ILC maps on small scales.We tweaked CMB-S4ʼs configuration to investigate if the residual CIB levels can be lowered further. To this end, we altered the noise level of bands in both CMB-S4 surveys by modifying the et al. (2015). Cluster detection sensitivity for CMB-S4 will be limited by residual CIB signals, while the confusion noise from diffuse tSZ is the dominant source of variance for CMB-HD. number of detectorsin each band. We do not find any improvement,which suggeststhat the current configuration listed in Table 1 (CMB-S4 Collaboration 2019) is the most optimal for the CMB-S4 cluster survey.
For CMB-HD, since the residual CIB level is expected to be much lower than CMB-S4 in our setup (Sehgal et al. 2019), one could expect the cluster sensitivity to be much higher than CMB-S4. However,the confusion noise from the diffuse tSZ (Holder et al. 2007) sets a noise floor hindering further improvementin sensitivity. Note that the gray signalband in Figure 2 is much higher than the noise curves N ℓ yy for CMB-HD in yellow. The tSZ confusion noise can be lowered by masking the detected clusters, but we defer a detailed investigation of this to a future work.
The sensitivity in regions of high galactic emission for S4-Wide is worse than the rest of the footprint by roughly 16% at all redshifts. For CMB-HD, the degradation is ∼28% for clusters with z 1 and ∼23% overall. While 30% S/N penalty is significant, we note thatit is an optimistic estimate given that our model for the galactic emission power spectrum (S0_d0 dust and S0_s0 from pySM3 simulations) is a simple power law. It ignores complexitieslike varying spectral or emissivity indices and non-Gaussianities, which can introduce non-negligible biases to the cluster tSZ signal. As a result,we do not consider the clusters in regions of high galactic emission for subsequent analyses in this work.
While not shown in Figure 3, in the absence of galactic emission, cluster limiting masses reduce by ∼7% compared to regions with low levels of galactic emission in the S4-Wide footprint. A significant fraction of this S/N degradation in the presenceof galactic emission is for nearby clusters, in agreementwith excess large-scale noise, green solid versus green dashed-dotted curves, in Figure 2. For all three surveys, the spike atlow redshift in Figure 3 is because we limit S/N calculation to 2 max q = ¢ . While 2
for clusters with z 0.5 and hence optimal, our choice of max q does notfully encompassthe cluster signal for nearby clustersand hence reduces their S/N.
The reason for the decrease in the minimum detectable mass with redshift is twofold. At low redshifts, the cluster S/N degrades because of residual contamination from atmospheric noise and CMB. At high redshifts, according to self-similar evolution of clusters (Kaiser 1986), a cluster with a given mass will have a higher temperature and hence a higher tSZ signal compared to its low redshift counterpart.This leads to an increase in cluster S/N when going from low to high redshifts.
Sensitivity can also be expressed in terms of cluster survey completeness as (Planck Collaboration et al. 2016b;Alonso et al. 2016) of clusters with a lower Y SZ c 500 signal.This is evident from the figure, where we note that curves move from lower to higher values of Y SZ c 500 for increasing levels of galactic emission (left to right). The slopes of individual lines also decrease in the same order. We note the same pattern when going from lownoise to high-noise surveys (CMB-HD → S4-Ultra deep → S4-Wide) and also from low to high cluster redshifts (blue to red). The redshifttrend is because of: (a) S/N degradation for low redshift clusters due to residual contamination from atmospheric noise and CMB and (b) S/N improvement due to selfsimilar evolution for higher redshift clusters. The significant S/ N penalty for lowest redshifts z 0.3 is due to the hard cutoff 2 max q = ¢ used for S/N calculation. See Section 3.2.1 for more discussion.
Based on these results, we find that S4-Wide shall detect (at 5σ) all galaxy clusters with an integrated Compton Y 10 ´-at z 1.5 over the delensing survey footprint ( f sky = 3%) shown in the bottom panel (G). The sensitivity of CMB-HD is roughly similar to S4-Ultra deep but over a large region f sky = 50% of sky as shown in panel (B).
We present a cumulative redshift distribution of clusters expected from the three surveysin Figure 5: CMB-HD in yellow, S4-Wide in green,and S4-Ultra deep in red.Cluster counts are obtained by sampling Tinkeret al. ( 2008) HMF using the MC sampling approach discussed in Section 2.8.1. Dashed-dotted lines for CMB-HD and S4-Wide correspond to clusters expected from regions with low galactic foregrounds. Solid curves are the total number of clustersfrom the full footprint, i.e., a combination of both low and high galactic emission regions, and the split between the two regions can be found in Table 3. Like in the previous Sections,galactic foregrounds are absent for S4-Ultra deep. For comparison, we show the currently available SZ cluster samples (S/N 4.5) from ACT (Hilton et al. 2018(Hilton et al. , 2021) ) as blue dashed, Planck (Planck Collaboration etal. 2016b) as red dotted, and SPT (Bleem et al. 2015(Bleem et al. , 2020;;Huang et al. 2020) as black dasheddotted curves.
S4-Wide shall detect close to 75,000 clusters in the baseline footprint ( f sky = 50%) while the S4-Ultra deep will obtain ∼14,000 clusters in CMB-S4ʼs de-lensing footprint ( f sky = 3%). While most of the low redshift z 1 clusters will be part of the LSST or eROSITA cluster samples (LSST Science Collaboration et al. 2009;Merloni et al. 2012), the redshift independent property of the tSZ signalwill open the unique high redshift discovery space for future CMB surveys. For example,S4-Wide (S4-Ultra deep)is expected to detect 1000 (350) clusters at z 2.
The number of clusters expected from CMB-HD is three times greater than that from S4-Wide. In the high redshift regime z 2,the expected numberfor CMB-HD is more than an order of magnitude higherthan CMB-S4. In Table 4, we give the median masses and redshifts of clustersin the baseline footprint from all three surveys. Average lensing mass estimates of the cluster sample using both temperature and polarization CMB-cluster lensing is also given in the table. While CMB temperature returns a higher lensing S/N for S4-Wide, we find a polarization channelto dominate the S/N for S4-Ultra deep. This is due to the higher noise floor set by foreground signals along with additional strategies used to mitigate foreground-induced bias in temperature-based lensing reconstruction. The same is true for CMB-HD but to a much lower extent since the variance from CIB is highly suppressed for CMB-HD (Sehgal et al. 2019). We report median mass and lensing estimates for both the full sample and also for clusters with z 2.
Modifying cluster virialization alters the cluster tSZ signal from clusters, which in turn affects the tSZ S/N. This is illustrated using the change in the minimum detectable cluster mass as a function of redshift in Figure 6 for S4-Wide with low levels of galactic emission.The thick solid black line is the baseline curve, the same as the green dashed-dotted curve in Figure 3. The thin dashed curves are for model 1 when we vary the virialization efficiency from η v ä [0.9, 1.1] based on Equations ( 11) and ( 12). As expected, the minimum detectable massesdecreasefor
The number of high redshift z 2 clusters from S4-Wide drop (increase) by two times for η v = 0.9(1.1) compared to 992 clusters for the fiducial value 1.0 v fid h = (see Table 3). The thick pink dashed-dotted curve is for model 2 based on Equation ( 13). It is similar to our baseline case (black), and we get roughly a 10% overallincrease in the number of clusters consistent with the trend in Figure 6.
Now we turn to parameter constraintsusing the Fisher formalism presented in Section 2.8. We combine N(z,M L , q) with primary CMB information from the three surveys to forecast standard errors on the parameters governing one of the two virialization models along with the Y M SZ -scaling relation (Equation ( 9)) and cosmological parameters.For cosmology,we focus on two-parameter extension to Lambda cold dark matter (ΛCDM) to include the sum of neutrino masses∑m ν and dark energy equation of state w 0 . Unless otherwise stated, cluster counts N(z, M L , q) in the rest of this section includes temperature-and polarization-based CMBcluster lensing mass calibration. The baseline redshift binning adopted was Δz = 0.1 for 0.1 z < 1.5 and one massive redshift bin for all high redshift clusters 1.5 z 3. A Plancklike prior has been assumed foroptical depth to reionization ( ) 0.007 re s t = . We also look into the following: (a) individual constraints from primary CMB and cluster counts, (b) importance of CMB-cluster lensing-based mass calibration, (c) impact of high redshift clusters and redshift binning, and (d) the effectof re t prior. These checks are limited to S4-Wide only.
In Figure 7, we presentthe marginalized constraints (68%
å n from all three surveys: CMB-HD in yellow, S4-Wide in green,and S4-Ultra deep in red. The lower and upper diagonals correspond to constraints for cluster virialization models 1 and 2, respectively.
The CMB-S4 and CMB-HD experiments can provide stringent constraintson the dark energy equation of state σ(w 0 ) and the sum of neutrino masses. We obtain 1%-2% on σ(w 0 ) from CMB-S4: 1.2% (1.8%) from S4-Wide (S4-Ultra deep) and 1% jointly from both CMB-S4 surveys. CMB-HD will provide sub-percent (0.5%) level constraints on w 0 . For neutrino masses, we obtain σ(∑m ν ) = 28 meV(45 meV)from S4-Wide (S4-Ultra deep), 23 meV jointly from both, and 13 meV from CMB-HD, enabling a ∼2.5σ-4.5σ detection of the sum of neutrino masses from both CMB-S4 and CMB-HD assuming a normal hierarchy lower limit of 60 meV. Although not shown in the figures, both experiments provide 1% constraintson the scalar fluctuation amplitude A s , Hubble parameter σ(h), and dark matter density σ(Ω c h 2 ). Adding largescale information from Planck has a negligible impact on the constraints from S4-Wide and CMB-HD, while it improves the cosmological constraints from S4-Ultra deep by 5%-10%.
Results for the first virialization model in Equation ( 11) are shown in the lower diagonal of Figure 7. We find that CMB-S4 clusters can help constrain σ(η v ) at the 2%-4% level, while CMB-HD can provide sub-percent level constraints. However, note that this assumes we have 100% knowledge about the astrophysics of low redshift clusters, which is not fully true but has been rapidly advancing (see recent review by Mroczkowski et al. 2019). Both experiments can provide sub-percent constraints on the HSE bias b 1 HSE -. While η v only modifies the tSZ signalof clusters atz 2 and is only constrained by them, low redshift clusters are also important in breaking the degeneraciesbetween other cosmological/scaling relation parameters and η v . For example,if we only consider clusters at z 2, η v is highly degenerate with parameters like b HSE or σ 8 , and adding low redshift information almost entirely breaks the degeneracies with other parameters.
The upper diagonal of Figure 7 shows the results forthe second cluster virialization model in Equation ( 13). In this case, we obtain σ(A v ) = 0.05 (0.1) from S4-Wide (S4-Ultra deep) for the redshift evolution parameter of the virialization, corresponding to ∼33% jointly from the two CMB-S4 surveys. For σ(B v ), we get 5% and 8% from the two CMB-S4 surveys and ∼4% jointly. CMB-HD will reduce the measurement uncertainty on these two parameters by more than three times.
Modifying cluster virialization from model 1 to model 2 does notintroduce statistically significant differences in other parameter constraints as can be seen by comparing the lower and upper diagonals in Figure 7.
The scatter in the Y M SZ -scaling relation is constrained to the 13%-16% level by CMB-S4, and to the 2% level by CMB-HD. The 1σ errors on mass and redshift evolution parameters of the relation are ( ) 0.01
for S4-Wide. When switching from model1 to model 2, we note a strong degeneracy between A v and Y g since they both probe the redshift evolution of the tSZ signal.The mass and redshift evolution parametersof the log-normal scatter(α σ and γ σ ) are an order of magnitude worse.The numbers are similar or sometimes slightly better for S4-Ultra deep,which is because of a better lensing S/N per cluster for S4-Ultra deep. We note that CMB-HD can improve constraints on the Y M SZ -scaling relation parameters by roughly an order of magnitude compared to CMB-S4.
In Figure 8, we present the constraints (68% CL) separately from CMB TT/EE/TE (orange dotted) and cluster counts (purple dashed) for S4-Wide. The green solid curves correspond to the joint CMB and cluster count constraints. As before,lower and upper diagonals correspond to virialization models 1 and 2. CMB spectra are insensitive to cluster virialization parameters
), and hence, the orange dotted curves are not shown for those parameters. However, CMB still helps in constraining them by breaking degeneracies with other parameters. That is the reason for the difference between the purple dashed (cluster counts) and green solid (joint constraints)curves. For other parameters, CMB spectra add minimal to modest levels of information compared to clusters. However, since CMB and cluster counts prefer Figure 7. Marginalized Fisher constraints (68% CL) obtained by combining information from primary CMB spectra (TT/EE/TE) and cluster counts N(z, M L , q): CMB-HD is in yellow, S4-Wide in green, and S4-Ultra deep in red. We use a Planck-like prior ( ) 0.007 re s t = for all surveys. Both surveys can reduce the uncertainty on the dark energy equation-of-state parameter σ(w 0 ) to 1%. Combining primary CMB with clusters will also enable ∼2.5σ-4.5σ detection of the neutrino masses. The lower and upper diagonals represent cluster virialization models 1 and 2, respectively.Model 1: Cluster virialization efficiency η v can be constrained to an accuracy of 2%-4% by S4-Wide and S4-Ultra deep, while CMB-HD can provide sub-percent level constraints. All surveys provide <1% constraints on the HSE bias parameter. Model 2: CMB-S4 can provide 33% and ∼4% constraints on A v and B v parameters, while CMB-HD reduces the uncertainties on both parameters by three times.Errors on other parameters do not change significantly between the two cluster virialization models. different nearly orthogonal degeneracy directions, the joint constraints offer remarkable improvements compared to either of them individually. For example,constraints on the sum of neutrino masses improve by ×2.5, from σ(∑m ν ) = 70 meV to 28 meV,when adding cluster counts to primary CMB spectra.
CMB-cluster lensing-based mass calibration is critical to obtain the results described above. To highlight the importance, we presentconstraints with (green solid)and without (pink dashed)CMB-cluster lensing information in Figure 9. When CMB-cluster lensing in excluded, we simply bin clusters in tSZ S/N q and redshift z: N(z, q). Since both cosmological and Y M SZ -scaling relation parameters affect the cluster redshift and tSZ S/N distributions, they can be constrained even in the absenceof lensing massesalbeit rather weakly (Louis & Alonso 2017). For example, errors on cosmological parameters σ(∑m ν ) = 60 meV and σ(w 0 ) = 0.03 both degrade by more than two times for pink without lensing compared to green curves with lensing mass calibration.Errors on virialization model parameters also degrade similarly by three times or more without lensing mass information.
The constraints presented abovehave been derived by binning clusters atz 1.5 in one massive high redshift bin. This is a conservativeapproach to take into account the difficulty of obtaining redshifts for distant clusters. We modify this choice using: case (i) an optimal setting with Δz = 0.1 for all clusters;case (ii) a less conservative setting with Δz = 0.1 for 0.1 z < 2 and Δz = 1 for 2 z 3; and case (iii) a pessimistic setting by ignoring clusters at z > 1.5.
Case (i): We note a ×3.8 better constraint on σ(η v ) = 0.0076 compared to the baseline case σ(η v ) = 0.0228 (see green curve in Figure 7). Constraints on other virialization model parameters b A B , , v v HSE also improve by 20%-30%. Similar improvements are seen for h, w 0 but this optimistic redshift binning scheme has a negligible (<10%) impact on A s , ∑m ν , and Ω c h 2 .
Case (ii): In this case,we see a threefold improvement on σ(η v ) = 0.0092 and a 15% improvement on b A B , , v v HSE compared to the baseline case, but this setting has a negligible (<10%) impact on other parameters. Case (iii): Since η v only affects clusters with z 2, we do not constrain η v with this pessimistic setting even though this is one Figure 8. Individual constraints (68% CL) from CMB TT/EE/TE spectra (orange dotted) and cluster counts (purple dashed) are shown. The combination of the two, our baseline setup, is shown as the green solid curves. The nearly orthogonal degeneracy directions of CMB spectra and cluster counts on structure growth param provide excellent joint constraints compared to either of them individually on σ(∑m ν ) and σ(w 0 ). Only S4-Wide is shown.
of the main goals of this work.Nevertheless we perform this test to addressthe challengesin obtaining redshifts and understanding the survey selection for clusters at z > 1.5. We note up to 15% degradation in constraints on cosmological parametersindicating that clusters with z 1.5 dominate cosmologicalconstraints.The other virialization model parameter constraints, b HSE and B v worsen by 15%, while A v , controlling the redshift evolution of model 2, degrades by more than 60%.
3.4.6.Effect of ( ) re s t Prior Here we check the effect of the Planck-like prior adopted in the forecasts above. Since a higher optical depth would suppresssmall-scaleCMB anisotropies, re t has significant correlation with parameters like h, Ω c h 2 , ∑m ν , and w 0 and the choice of ( ) (Hazra et al. 2018;Di Valentino et al. 2018). Even though this has an effect on CMB-only constraints on all of the parameters listed above, we note significant effects only on ∑m ν and Ω c h 2 with the joint CMB and cluster count information.
Removing re t prior degrades σ(∑m ν ) by ×1.5. With ( ) 0.002 re s t = , σ(∑m ν ) improves by ×1.3-1.5 from the three surveys. We obtain a similar level of changes on σ(Ω c h 2 ) with the two settings.The prior on re t does notaffect virialization model parameter constraints.
Thus far, the forecasted clusternumber counts and the cosmologicalconstraintsare obtained using noise levels in Table 1 expected to be achieved at the end of survey periods. Given that S4-Wide is expected to start operations close to the end of this decade, these constraints may not be achieved until Figure 9. Importance of CMB-cluster lensing mass calibration for cluster counts. The green curves are the same in the left panels, while the pink dashed curves are constraints obtained without CMB-cluster lensing information. Ignoring lensing mass calibration degrades the constraints significantly for all parameters. Ellipses ar 68% CL regions,and only S4-Wide is shown. the middle of next decade.In this section, we check the dependence of cluster counts and cosmological constraints as a function of observing time for S4-Wide. For this test we simply scale the noise levels in each band by N N years baseline where N baseline = 7 yr. We go from 1 yr to 10 yr. Note that this simple scaling assumesfull deploymentat the start of the survey, which could be unrealistic, and the noise scaling may be slightly more complicated for the first few years in reality.
In Figure 10, we show the cumulative clustercounts for clusters above multiple redshifts as a function of the number of S4-Wide observation years. It is evident from the figure that the S4-Wide clustersample,with ∼20,000 clusters, will surpass the currentSZ samples from ACT,Planck,and SPT (purple dotted line) even at the end of the first year of observation. In fact, at the end of the first year, the S4-Wide sample will have close to 5000 clusters at z 1, similar to the total number of current SZ clusters at all redshifts. It is also worth noting that we use an S/N threshold of 5 for S4-Wide, while the current SZ samples shown here use 4.5.
The number of clusters expected at the end of each year cannot be obtained from baseline results (7 yr) using a simple noise scaling because of residual foregrounds in the Compton-y maps. Since the residualforegrounds(mostly CIB) dominate small scales, the impact of residual foregrounds is more important for high redshift clusters that span a smaller angular extent on the sky. For example,scaling S4-Wide clusters at z 2 from
= clusters while we have 200 clusters (two times lower) at the end of year 1 in Figure 10.
Cosmological constraints from clusters and primary CMB as a function of S4-Wide observation yearsare presented in Figure 11. Similar to cluster forecasts, noise levels in each band are scaled from Table 1 for each year to obtain the CMB Fisher matrix. The improvementin constraints is not dramatic as a function of observing years, and this is primarily because of different degeneracy directions in the parameter space probed by clusters and primary CMB. However, note that cluster cosmology is not the only science driver for CMB-S4. It has a broad range of science goals including the measurement of light relic density and the production legacy catalogs in millimeter/submillimeter wavelengthsthat require the proposed N baseline = 7 yr (CMB-S4 Collaboration 2019) to produce wide and deep CMB maps. With ∼200 (400) clusters at z 2 at the end of year 1 (3), we find that S4-Wide can make a giant leap toward understanding the astrophysics and the onsetof the virialization mechanism of high redshift clusters, which are completely unconstrained currently.
In our baseline approach,we ignored cluster correlated foreground signals, namely the cluster kSZ signal and emission from DSFGs and RGs within clusters.For clusters close to detection limits for all three surveys considered here, cluster kSZ signals are less important as they are expected to be much smaller than the tSZ signal. Moreover, kSZ can be both positive or negative depending on the direction of the radial motion and hence only acts as an additional source of variance in our analysis.On the other hand, emission from DSFG and radio galaxies, since they are always positive, can fill in the tSZ decrementsthereby potentially contaminating tSZ measurements. While the presence of DSFG signals within clusters has been identified in Planck clusters (Planck Collaboration et al. 2016d), Melin et al. (2018a) reported that signals from DSFGs degrade the completeness of the Planck cluster catalog by 10% and showed thatthis contamination has a negligible impact on cosmological parameterinference.However, the Planck cluster sample deals with massive low redshift clusters where the star formation has been observed to be highly suppressed (Popesso et al. 2015). Furthermore, the cluster tSZ signal goes as M 5/3 while DSFG signals are roughly linear. As a result,DSFG contamination may be insignificant for Planck clusters. In this work, we are particularly interested in low-mass (M 500c 10 14 M e ) and high redshift z 1.5 clusters, near the peak of cosmic star formation history, to constrain cluster Figure 10. Cumulative clustercounts for S4-Wide shown as a function of number of observation years. The number of clusters in a 1 yr S4-Wide sample will surpass the currently available SZ cluster samples (purple dashed-dotted line). The counts cannot be simply scaled from the baseline observing period of 7 yr because ofresidualforeground signals that dominate smallscales and hence have a major impact on high-z clusters. For example,the actualz 2 clusters atthe end of the first year are two times lower than what will be obtained using a simple noise scaling.
Figure 11. Relative cosmological constraints be obtained by combining clusters with primary CMB as a function of S4-Wide observing years.Our results indicate that S4-Wide can return remarkable cosmological constraints compared to current limits even in its first few years of observation. S4-Wide, at the end of the first few years of observations, can also place compelling constraints on the virialization mechanism of high redshift that is currently unconstrained. astrophysics; hence,dust contamination mightpotentially be important.
We check the impact of cluster correlated signals using Websky (Stein et al. 2020) 2021). No such scaling was applied to MDPL2 simulations. In both cases, we pick 100 cutouts for every point in the M 500c , z grid. Due to the availability of a single Websky/MDPL2 mock sky realization,the number of kSZ/CIB signals available for this test reduces significantly for clusters with mass M 500c 3 × 10foot_3 M e at z 2.Hence, we limit this test to clustersbelow this mass and redshift range. We inject the cluster correlated signals from Websky/MDPL2 into our simulations in all of the frequency bands along with cluster tSZ signal, experimental noise,CMB and other astrophysical foregrounds described in Section 2.1, which are then passed through the ILC pipeline.
For high redshift (z 1) clusters near the detection limit, we note that DSFGs within clusters shift the tSZ-based cluster massesslightly lower. However, the bias is subdominant compared to statistical uncertainties at roughly the 0.2σ level for all three surveys. The bias is almostzero for low redshift clusters.As expected,cluster kSZ signals have a negligible impact on the recovered tSZ signals.
Both Websky and MDPL2 do not contain emission from RGs within clusters. Subsequently, we choose an extremely conservative test to assess the impact of RGs within clusters on the recovered tSZ signals. We inject a constant flux of S 150 = 0.1 mJy or 0.5 mJy for all clusterswhere the latter roughly matches the point source sensitivity (3σ) at 90 GHz for the CMB-S4 survey (CMB-S4 Collaboration 2019). The signal is scaled to other bands assuming a spectral index α radio = -0.6 (Everett et al. 2020). This test is limited to S4-Wide only. We find that an S 150 = 0.1 mJy (0.5 mJy) can bias tSZ measurements low by 0.1-0.2σ (1σ). While a 1σ systematic error is large, we note that our model for RGs is unrealistic, and hence our results should only be interpreted as an upper limit of the systematic error.
Our simple RG model can be potentially replaced using the Websky simulations,which are currently being upgraded to include RG signals correlated with the underlying dark matter.
Similarly the limitation due to the smaller number of Websky or MDPL2 haloes at the high-mass end can be addressed using multiple realizations of the millimeter-wave sky,as released recently by Han etal. (2021) using deep learning techniques, for example. We leave these detailed studies for a future work.
Given that point sources in the maps can be misclassified as clusters, we check the effect of point sources using 100 noiseonly simulations. The simulations for this test include astrophysical foregrounds, CMB, experimental noise and point source signals. Cluster tSZ signalis ignored here.We model the point source signals in three different ways.
In the first case, we add the cluster correlated DSFG signals in each mass and redshiftbin using Websky simulations as explained above.We obtain zero false detections,which is consistent with a negligible systematic bias from dusty sources estimated above in Section 3.6.In a similar spirit, we also check the effectof random pointsources in the maps. In this case, we inject point source signals with fluxes S 150 ä [0.5, 1, 2, 3] mJy, which are then scaled to other bands using a power-law relation with a spectral index α dust = 3.2 and α radio = -0.6 to represent DSFG and radio pointsource signals (George et al. 2015). Again, we do not see any false detections from DSGFs. This indicates thatDSFGs do not show up as 5σ positive peaks in the Compton-y maps. For radio point sources, we find that the sourceswith flux S 150 2 mJy can be potentially problematic.However,these radio pointsources show up as negative peaks in the Compton-y maps and hence will not be classified as clusters.
We forecasted the numberof galaxy clusters thatcan be detected using future CMB surveys, namely S4-Wide, S4-Ultra deep,and CMB-HD. Our forecasts used realistic simulations that include signals from galactic and extragalactic signals along with atmospheric and instrumental noise components. In the baseline footprint with f sky = 50%, S4-Wide can detect close to 75,000 clusters, and the CMB-HD sample will contain three times more clusters. The smaller but deeper S4-Ultra deep survey can detect ∼14,000 clusters. Of these, 6000 (1500) will be at z 1.5, and 1000 (350) clusters will be at z 2 in the S4-Wide (S4-Ultra deep) cluster sample.The number of z 2 clustersis an order of magnitude higherfor the CMB-HD experiment. Including regions close to the galactic plane ( f sky = 17%) increases the sample size by roughly 20%.
The residual foreground signals, CIB in particular, dominate the small-scale variance in the Compton-y maps for CMB-S4. For CMB-HD, the variance from CIB is ×17 lower at 150 GHz due to efficient subtraction of dusty galaxy sources (Sehgal et al. 2019). The CIB subtraction and a five-times-smaller beam are the reasons for a much larger cluster sample from CMB-HD. Given the importance of residual CIB signals and the tSZ × CIB correlation, we checked the systematic errors in the recovered clustertSZ signals due to emission from galaxies within clusters.The effectof dusty star-forming galaxies was studied using Websky/MDPL2 simulations,while for radio galaxies,we use a simple constant flux model for all clusters. Our results indicate that systematic error due to the presence of dusty galaxies is much smaller than the statistical error 0.2σ, but having a constant radio galaxy signal with S 150 = 0.5 mJy can introduce ∼1σ bias. The models used for galactic foregrounds and RGs are basic and must be extended further. Nevertheless, the tests we performed to assess their contamination on the recovered tSZ signals are important for future SZ surveys.
We used a CMB-clusterlensing signalfrom both temperature and polarization to calibrate the cluster tSZ-mass scaling relation. We have ignored weak-lensing information from optical surveys in this work and note that including them can further improve the constraining poweras well as act as an important systematiccheck for CMB-cluster lensing-based mass estimates. The internally calibrated cluster counts were combined with primary CMB (TT/EE/TE) spectra to derive cosmologicalconstraints assuming a two-parameter extension to the standard model of cosmology (ΛCDM + ∑m ν + w 0 ). We show that the constraints on the dark energy equation-of-state σ(w 0 ) parameter can be between 1% and 2% for S4-Wide/S4-Ultra deep and sub-percent for CMB-HD. Similarly, the sum of neutrino masses∑m ν can detected at 2.5σ-4.5σ by both CMB-S4 and CMB-HD surveys assuming a normal hierarchy lower limit of 60 meV. We also assessthe importance of combining cluster counts with primary CMB,significance of CMB-cluster lensing, choice of ( ) re s t prior, effect of different redshift binning, and dependence ofour result on the total survey time.
In addition to cosmology and scaling relation constraints, we also study the evolution of the ICM using two models. In the first case, we model cluster virialization in Equation ( 11) using the standard HSE bias parameter and a virialization efficiency parameter η v that linearly scales with the tSZ signal from high redshift z 2 clusters. We find σ(η v ) = 0.00471,0.0228,and 0.0339 from CMB-HD, S4-Wide,and S4-Ultra deep, respectively, indicating that the mean deviation in thermal energy of z 2 clustersfrom their low redshift counterpartscan be constrained to roughly 2%-4% by CMB-S4 and <1% by CMB-HD experiments. All three surveys can place sub-percent constraints on the HSE bias parameter. Our second model in Equation ( 13) is more physically motivated and calibrated using Omega500 hydrodynamical cosmological simulations. In this case,CMB-S4 can provide ∼4% constraints on B v and ∼33% on A v , which controls the redshift evolution of the virialization mechanism. CMB-HD improves the constraining power by more than three times. This work represents a key step toward understandingthe selection function of high redshift clusters and the evolution of the ICM using current and future CMB SZ surveys like AdvACT (Henderson et al. 2016), CMB-HD (Sehgal et al. 2019), CMB-S4 (CMB-S4 Collaboration 2019), SPT-3G (Benson et al. 2014;Bender et al. 2018), and SO (Ade et al.2019).The binned cluster counts N(z, M L , q), Fisher matrices, and other associated productscan be downloaded. 16Appendix A Physically Motivated Cluster Virialization Model Galaxy clusters are dynamically active objects and generally out of HSE due to mergers and mass accretion processes. The lack of virialization is characterized by a nonthermal pressure fraction, P th /P tot , which quantifies the fraction of energy densities in unvirialized bulk and turbulent gas motions compared to the total pressure,P tot = P th + P nth (e.g., Lau et al. 2009;Battaglia etal. 2012;Nelson etal. 2014a;Shi & Komatsu 2014;Yu et al. 2015). In the presenceof the nonthermal pressure, the total pressure is given by the sum of the thermaland nonthermal pressure: P tot = P th + P nth , which in turn provides pressure support against gravitational collapse. 9), and Y is the integrated pressure of each component within the sphere of R 500c following Equation (6). Note that the nonthermalpressure is one of the dominant sources of systematic uncertainties in the HSE mass bias (e.g., Nagai et al.2007b;Lau et al.2013;Shi et al. 2016;Biffi et al. 2016;Angelinelli et al.2020).
We compute the impact of the nonthermalpressure on the Y M SZ -relation of high redshift clusters using the model presented in Green et al. (2020). First, we use the analytical model of Shi & Komatsu (2014) is the total velocity dispersion,and η is the fraction of energy accreted thatis injected into turbulence motion. Due to the cosmic mass accretion process, the total velocity dispersion increases over time. The turbulence decaysinto thermal energy over the dissipation timescale t dis , which is proportional to the eddy turn-over time of the largest eddies, which is in turn proportionalto the local orbital time, t dis (r) = βt orb (r)/2. The model has been calibrated using Omega500 hydrodynamical cosmologicalsimulations (Nelson etal. 2014b), yielding the best-fit parameters of β = 1 and η = 0.7 (Shi et al. 2015). Given a cluster with mass M 500c , we generatethe averagemass accretion history M(t) and concentration c(t) of the cluster following van den Bosch et al. (2014) and Zhao et al. (2009), respectively.For the total pressure profiles, we use the KS01 (Komatsu & Seljak 2001) model, which is based on a polytropic gas in HSE with the NFW profile.
Figure A1 shows the fraction of the Y( < R 500c ) signal in nonthermalpressure, Y nth /Y tot as a function of redshift.For a constantmass M 500c = 10 14 M e /h, the modelpredicts thatthe fraction of the Y signal in nonthermal pressure can evolve from 20% at z = 0 to 40% at z = 3, indicating strong redshift dependence. The model predicts enhancement in the nonthermal pressurefraction toward high redshift due to the enhanced mass accretion rate in the early universe (Green et al. 2020). Our result suggests thatthe evolution of Y nth /Y tot at M 500c = 10 14 M e /h can be described by a simple function:
where A v and B v are calibrated to 0.155 and 0.189.
Figure A1. The ratio of the nonthermal pressure fraction, Y nth /Y tot , enclosed within the projected aperture radius of R 500c for an M 500c = 10 14 M e /h galaxy cluster as a function of redshift z. The model (orange) predicts that the nonthermalpressure increases as a function of redshift due to the enhanced mass accretion rate in the early universe. We propose a fitting function (black solid), Equation (A3),which describes the redshift evolution out to z ≈ 3.
The AstrophysicalJournal, 926:172 (19pp), 2022 February 20 Raghunathan et al.
S4-Ultra deep is also expected to have a 20 GHz band, but we ignore that in this work for simplicity.
https://github.com/CMB-S4/s4mapbasedsims/tree/master/202102_ design_tool_input
https://mocks.cita.utoronto.ca/websky
http://behroozi.users.hpc.arizona.edu/MDPL2/hlists/
https://github.com/sriniraghunathan/tSZ_cluster_forecasts/
https://people.cmb-s4.org/public/showdir.php